`hierarchy_very_small`

proved

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module: IndisputableMonolith.QFT.UVCutoff
domain: QFT
line: 189 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.QFT.UVCutoff on GitHub at line 189.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

formal source

 186def hierarchyRatio : ℚ := higgsMassGeV / planckMassGeV
 187
 188/-- **THEOREM**: The hierarchy ratio is extremely small (~10⁻¹⁷). -/
 189theorem hierarchy_very_small :
 190    hierarchyRatio < 1 / 10^16 := by
 191  unfold hierarchyRatio higgsMassGeV planckMassGeV
 192  norm_num
 193
 194/-! ## Predictions and Tests -/
 195
 196/-- Testable predictions from RS UV cutoff:
 197
 198    1. **No trans-Planckian modes**: Physics is regular at Planck scale
 199    2. **Modified dispersion relations**: E² = p² + m² gets corrections at high p
 200    3. **Cosmic ray spectrum**: GZK cutoff might be modified
 201    4. **Black hole formation**: Minimum mass related to τ₀
 202    5. **Loop corrections**: Finite and calculable with RS cutoff -/
 203def predictions : List String := [
 204  "Dispersion relation corrections at E ~ E0",
 205  "Modified loop corrections near cutoff",
 206  "Finite quantum gravity effects",
 207  "Discrete spacetime effects in cosmology"
 208]
 209
 210/-! ## Comparison with Other Approaches -/
 211
 212/-- Other UV regularization approaches:
 213
 214    | Approach | Cutoff | Physical? |
 215    |----------|--------|-----------|
 216    | Dimensional reg | ε → 0 | No |
 217    | Pauli-Villars | Heavy mass M | Artificial |
 218    | Lattice QCD | Lattice spacing a | Physical on lattice |
 219    | String theory | String length l_s | Yes |

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